Isn’t a gas turbine technically an external combustion engine?
The Sterling engine is the best-known external combustion engine.
This makes sense, but the motor needs to be big enough to keep the batteries fully charged to be really useful. This is How Diesel Locomotives Work.
For trains, the main advantage seems to be easier power transmission to the wheels. This doesn’t help as much in a car, but I can still see some advantages. The engine can be smaller because it does not need to be big enough to accelerate the car, the electric motors do that. The engine can be designed to operate at a very narrow rpm range, making it much more efficient. The engine could be designed to shut down when the batteries are full. The transmission could be eliminated. These advantages need to be weighed against the power losses from transferring all the power to electricity. I assume the auto makers have decided it does not work out, but it seems like something that could be explored.
I don’t think you’re answering the OP’s question. You’re looking at the tradeoffs between parallel and series hybrid electric vehicles.
The OP is asking about the tradeoffs between different strategies of series hybrid operation: 1) all-electric operation for X miles followed by charge-sustaining engine operation, versus 2) continuous engine operation, using an engine that isn’t large enough to keep the batteries charged.
Note that the Chevy Volt is a series (plug-in) hybrid.
One other advantage with trains is that it’s very easy to use the same locomotives off of overhead catenary wires, instead of the diesel engine. Large portions of the rail network on the East Coast are wired, presumably either because electric power is cheaper, or to cut down on pollution from the diesels (or a combination of both). Last I knew, the Pennsylvania main line switches over from catenary to diesel at Harrisburg, and it only takes about five minutes to switch it (they don’t need to disconnect any cars from the locomotive, or the like).